Consider a real Gaussian stationary process, either on Z or on R. That is,
a stochastic process, invariant under translations, whose finite marginals
are centered multi-variate Gaussians. The persistence of such a process on
[0,N] is the probability that it remains positive throughout this interval.
The relation between the decay of the persistence as N tends to infinity
and the covariance function of the process has been investigated since the
1950s with motivations stemming from probability, engineering and
mathematical physics. Nonetheless, until recently, good estimates were
known only for particular cases, or when the covariance kernel of the
process is either non-negative or summable.
In the talk we discuss a new spectral point of view on persistence which
greatly simplifies its analysis. This is then used to analyze the
qualitative behavior of the persistence probability in a very general
setting.
Based on joint work with Ohad Feldheim and Shahaf Nitzan.